Let {A, B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to （A, B） if it is disjoint from A U B and it cuts F into two once-punctured tor...Let {A, B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to （A, B） if it is disjoint from A U B and it cuts F into two once-punctured tori X, Y with A belong X, B belong Y. Let γ be a simple closed curve on F which is disjoint from A ∪ B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hγ^n（C） ： n ∈ Z} is the set containing all the simple closed curves on F which is separating with respect to （A, B）, where hγ is the Dehn twist along γ on F. This also shows how two simple closed curves on F which are separating with respect to （A, B） are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.展开更多
This review summarizes the recent advances on the application of ^(57)Fe Mössbauer spectrometry to study the magnetic and phase characteristics of Nd–Fe–B-based permanent magnets. First of all, the hyperfine st...This review summarizes the recent advances on the application of ^(57)Fe Mössbauer spectrometry to study the magnetic and phase characteristics of Nd–Fe–B-based permanent magnets. First of all, the hyperfine structures of the Ce_(2)Fe_(14)B,(Ce,Nd)_(2)Fe_(14)B and MM_(2)Fe_(14)B phases are well-defined by using the model based on the Wigner-Seitz analysis of the crystal structure. The results show that the isomer shift δ and the quadrupole splitting öEQ of those 2:14:1 phases show minor changes with the Nd content, while the hyperfine field Bhfincreases monotonically with increasing Nd content and its value is influenced by the element segregation and phase separation in the 2:14:1 phase. Then, the hyperfine structures of the low fraction secondary phases are determined by the ^(57)Fe Mössbauer spectrometry due to its high sensitivity. On this basis,the content, magnetic behavior, and magnetization of the REFe_(2) phase, the amorphous grain boundary(GB) phase, and the amorphous worm-like phase, as well as their effects on the magnetic properties, are systematically studied.展开更多
基金The NSF (10171024,10471020) of Chinaa grant of the outstanding Youth fellowship of Hei Long Jiang Province and the Science Foundation (20060106) for Yong Teachers of Northeast Normal University.
文摘Let {A, B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to （A, B） if it is disjoint from A U B and it cuts F into two once-punctured tori X, Y with A belong X, B belong Y. Let γ be a simple closed curve on F which is disjoint from A ∪ B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hγ^n（C） ： n ∈ Z} is the set containing all the simple closed curves on F which is separating with respect to （A, B）, where hγ is the Dehn twist along γ on F. This also shows how two simple closed curves on F which are separating with respect to （A, B） are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.
基金Project supported by the Zhejiang Provincial Natural Science Foundation of China(Grant Nos.LY20E010002,LR18E010001,and LD19E010001)the National Natural Science Foundation of China(Grant Nos.U1704253 and 51801047)+2 种基金Liaoning Revitalization Talents Program(Grant No.XLYC1807177)the Key Research and Development plan of Zhejiang Province,China(Grant No.2020C05014)Ten Thousand Talents Plan of Zhejiang Province,China(Grant No.2018R52003)。
文摘This review summarizes the recent advances on the application of ^(57)Fe Mössbauer spectrometry to study the magnetic and phase characteristics of Nd–Fe–B-based permanent magnets. First of all, the hyperfine structures of the Ce_(2)Fe_(14)B,(Ce,Nd)_(2)Fe_(14)B and MM_(2)Fe_(14)B phases are well-defined by using the model based on the Wigner-Seitz analysis of the crystal structure. The results show that the isomer shift δ and the quadrupole splitting öEQ of those 2:14:1 phases show minor changes with the Nd content, while the hyperfine field Bhfincreases monotonically with increasing Nd content and its value is influenced by the element segregation and phase separation in the 2:14:1 phase. Then, the hyperfine structures of the low fraction secondary phases are determined by the ^(57)Fe Mössbauer spectrometry due to its high sensitivity. On this basis,the content, magnetic behavior, and magnetization of the REFe_(2) phase, the amorphous grain boundary(GB) phase, and the amorphous worm-like phase, as well as their effects on the magnetic properties, are systematically studied.
